Non-linear performance measures are widely used for the evaluation of learning algorithms. For example, $F$-measure is a commonly used performance measure for classification problems in machine learning and information retrieval community. We study the theoretical properties of a subset of non-linear performance measures called pseudo-linear performance measures which includes $F$-measure, \emph{Jaccard Index}, among many others. We establish that many notions of $F$-measures and \emph{Jaccard Index} are pseudo-linear functions of the per-class false negatives and false positives for binary, multiclass and multilabel classification. Based on this observation, we present a general reduction of such performance measure optimization problem to cost-sensitive classification problem with unknown costs. We then propose an algorithm with provable guarantees to obtain an approximately optimal classifier for the $F$-measure by solving a series of cost-sensitive classification problems. The strength of our analysis is to be valid on any dataset and any class of classifiers, extending the existing theoretical results on pseudo-linear measures, which are asymptotic in nature. We also establish the multi-objective nature of the $F$-score maximization problem by linking the algorithm with the weighted-sum approach used in multi-objective optimization. We present numerical experiments to illustrate the relative importance of cost asymmetry and thresholding when learning linear classifiers on various $F$-measure optimization tasks.

研究了一些非线性性能度量的理论属性，包括F-度量，Jaccard指数等，将这些性能度量优化问题归约为具有未知成本的成本敏感分类问题，并提出了一种具有可证明保证的算法来获得近似最优分类器以解决F-度量优化问题。

优化拟线性性能度量的理论：以F-度量为例